Induction, Mutual Inductance, and Magnetic Field Energy

Vocabulary flashcards covering mutual inductance, self-inductance, inductors, magnetic energy storage, and related mathematical formulations.

Inductance

The property of a device that describes how effectively it induces an emf in another device, expressing the effectiveness of a given physical arrangement.

Mutual Inductance ($M$)

A geometric quantity independent of current that relates the emf induced in one circuit to the rate of change of current in a nearby circuit, defined by $M = \frac{N_2\Phi_{21}}{I_1} = \frac{N_1\Phi_{12}}{I_2}$.

Henry ($\text{H}$)

The SI unit of mutual inductance and self-inductance, named after Joseph Henry, equivalent to $1\,\text{H} = 1\,\text{V}\cdot\text{s/A}$.

Mutually Induced EMF

An electromotive force produced in a circuit by the time-varying current in an adjacent circuit, expressed as $\varepsilon_2 = -M\frac{dI_1}{dt}$ and $\varepsilon_1 = -M\frac{dI_2}{dt}$.

Counter-Wound Coils

A coil winding configuration where neighboring coils carry current in opposite senses so their magnetic fields cancel, significantly reducing mutual inductance with surrounding conductors or casings.

Self-Inductance ($L$)

The constant of proportionality relating the magnetic flux through a circuit to the current producing that flux within the circuit itself, defined by $N\Phi_m = LI$.

Self-Induced EMF

The electromotive force produced across an inductor due to a changing current passing through it, given by $\varepsilon = -L\frac{dI}{dt}$, whose polarity opposes the change in current.

Inductor

A circuit element designed specifically to provide self-inductance in a circuit, typically consisting of a relatively long coil of wire.

Self-Inductance of a Cylindrical Solenoid

The self-inductance of an ideal solenoid of length $l$, cross-sectional area $A$, and $N$ turns, given by $L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 A l = \mu_0 n^2 V$, where $n = \frac{N}{l}$ and $V = A l$.

Self-Inductance of a Rectangular Toroid

The self-inductance of a toroid with a rectangular cross-section of height $h$, inner radius $R_1$, outer radius $R_2$, and $N$ turns, given by $L = \frac{\mu_0 N^2 h}{2\pi}\ln\left(\frac{R_2}{R_1}\right)$.

Magnetic Energy Density ($u_m$)

The energy stored per unit volume in a magnetic field, given by the expression $u_m = \frac{B^2}{2\mu_0}$.

Energy Stored in an Inductor ($U$)

The total work required to establish current $I$ and stored in the inductor's magnetic field, given by $U = \frac{1}{2}LI^2$.

Power Absorbed by an Inductor ($P$)

The instantaneous rate of energy transfer into an inductor when carrying instantaneous current $i$, defined as $P = \varepsilon i = L\frac{di}{dt}i$.

Oscillation

The fluctuation of a quantity, or repeated regular fluctuations of a quantity, between two extreme values around an average value.